Related papers: Optimal l-one Rank One Matrix Decompositions
We will use commutators to provide decompositions of $3\times 3$ matrices as sums whose terms satisfy some polynomial identities, and we apply them to bounded linear operators and endomorphisms of free modules of infinite rank. In…
In this paper, we describe a low-rank matrix completion method based on matrix decomposition. An incomplete matrix is decomposed into submatrices which are filled with a proposed trimming step and then are recombined to form a low-rank…
We develop tractable convex relaxations for rank-constrained quadratic optimization problems over $n \times m$ matrices, a setting for which tractable relaxations are typically only available when the objective or constraints admit spectral…
We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP. The optimization problems…
Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results from the theory of matroids by establishing connections to networks by decomposition…
Motivated by Hirano-Tominaga's work \cite{HT} on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results \cite{de} on decomposing every matrix over a field of positive characteristic as a sum of…
Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone…
Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal…
One of the main issues in computing a tensor decomposition is how to choose the number of rank-one components, since there is no finite algorithms for determining the rank of a tensor. A commonly used approach for this purpose is to find a…
The idea of decomposing a matrix into a product of structured matrices such as triangular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions,…
In this paper we examine a symmetric tensor decomposition problem, the Gramian decomposition, posed as a rank minimization problem. We study the relaxation of the problem and consider cases when the relaxed solution is a solution to the…
In this paper, the canonical polyadic (CP) decomposition of tensors that corresponds to matrix multiplications is studied. Finding the rank of these tensors and computing the decompositions is a fundamental problem of algebraic complexity…
We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative…
It is shown that each positive map between matrix algebras is the sum of a maximal decomposable map and an atomic map which is both optimal and co-optimal. The result is analyzed in detail for the positive projection onto a spin factor.
We consider wreath product decompositions for semigroups of triangular matrices. We exhibit an explicit wreath product decomposition for the semigroup of all n-by-n upper triangular matrices over a given field k, in terms of aperiodic…
This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix…
In polarization optics, an important role play Mueller matrices -- real four-dimensional matrices which describe the effect of action of optical elements on the polarization state of the light, described by 4-dimensional Stokes vectors. An…
We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix $A$ is completely positive. Our decomposition can be applied to a wide range of matrices. We give alternate proofs for a number of related results…
We study the closure of the projection of the (nonconvex) cone of rank restricted positive semidefinite matrices onto subsets of the matrix entries. This defines the feasible sets for semidefinite completion problems with restrictions on…
We study rank-1 {L1-norm-based TUCKER2} (L1-TUCKER2) decomposition of 3-way tensors, treated as a collection of $N$ $D \times M$ matrices that are to be jointly decomposed. Our contributions are as follows. i) We prove that the problem is…